Unconditional Error Analysis of VEMs for a Generalized Nonlinear Schrödinger Equation (Journal of Computational Mathematics, 2024)
In this work, we focus on the conforming and nonconforming leap-frog virtual elementmethods for the generalized nonlinear Schrödinger equation, and establish their unconditional stability and optimal error estimates. By constructing a time-discrete system, theerror between the solutions of the continuous model and the numerical scheme is separatedinto the temporal error and the spatial error, which makes the spatial error $τ$-independent.The inverse inequalities in the existing conforming and new constructed nonconformingvirtual element spaces are utilized to derive the $L^∞$-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio, and then unconditionallyoptimal error estimates of the numerical schemes are obtained naturally. What needs tobe emphasized is that if we use the pre-existing nonconforming virtual elements, there isno way to derive the $L^∞$-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally, several numerical examples are reported to confirm our theoretical results.
Keywords: Conforming and nonconforming; Virtual element methods; Leap-frog scheme; Generalized nonlinear Schrödinger system; Unconditionally optimal error estimates.